Problem: Simplify; express your answer in exponential form. Assume $y\neq 0, p\neq 0$. $\dfrac{{(y^{-5}p^{5})^{5}}}{{(y^{-4}p^{3})^{-2}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-5}p^{5})^{5} = (y^{-5})^{5}(p^{5})^{5}}$ On the left, we have ${y^{-5}}$ to the exponent ${5}$ . Now ${-5 \times 5 = -25}$ , so ${(y^{-5})^{5} = y^{-25}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-5}p^{5})^{5}}}{{(y^{-4}p^{3})^{-2}}} = \dfrac{{y^{-25}p^{25}}}{{y^{8}p^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{-25}p^{25}}}{{y^{8}p^{-6}}} = \dfrac{{y^{-25}}}{{y^{8}}} \cdot \dfrac{{p^{25}}}{{p^{-6}}} = y^{{-25} - {8}} \cdot p^{{25} - {(-6)}} = y^{-33}p^{31}$